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High strain-rate tensile testing and viscoplastic parameter identification usingmicroscopic high-speed photography

Kajberg, J.; Sundin, K. G.; Melin, L. G.; Ståhle, P.

Published in:International Journal of Plasticity

DOI:10.1016/S0749-6419(03)00041-X

2004

Document Version:Peer reviewed version (aka post-print)

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Citation for published version (APA):Kajberg, J., Sundin, K. G., Melin, L. G., & Ståhle, P. (2004). High strain-rate tensile testing and viscoplasticparameter identification using microscopic high-speed photography. International Journal of Plasticity, 20(4-5),561-575. https://doi.org/10.1016/S0749-6419(03)00041-X

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High strain-rate tensile testing and viscoplasticparameter identification using microscopic

high-speed photography

J. Kajberga, K.G. Sundina,*, L.G. Melinb,1, P. Stahlea,2

aDivision of Solid Mechanics, Lulea University of Technology, SE-97187 Lulea, SwedenbDivision of Experimental Mechanics, Lulea University of Technology, SE-97187 Lulea, Sweden

Received in revised form 4 February 2003

Abstract

A combined experimental/numerical method for determination of constitutive parametersin high strain-rate material models is presented. Impact loading, using moderate projectilevelocities in combination with small specimens (sub mm) facilitate tensional strain rates in the

order of 104–105 s�1. Loading force is measured from one-dimensional wave propagation in arod using strain gauges and deformation is monitored with a high-speed camera equippedwith a microscope lens. A sequence of digital photographs is taken during the impact loading

and the plastic deformation history of the specimen is quantified from the photographicrecord. Estimation of material parameters is performed through so called inverse modelling inwhich results from repeated FE-simulations are compared with experimental results and abest choice of constitutive parameters is extracted through an iterative optimisation procedure

using the simplex method. Results are presented from a preliminary tension test of a mild steel(A533B) at a strain rate well over 104 s�1. The sensitivity of the evaluated material parametersto errors in measured quantities is studied. The method, especially the optical technique for

measurement of deformation will be further developed.# 2003 Elsevier Ltd. All rights reserved.

Keywords: B. Constitutive behaviour; B. Viscoplastic material; C. Impact testing; C. Optimization;

High-speed photography

International Journal of Plasticity 20 (2004) 561–575

www.elsevier.com/locate/ijplas

0749-6419/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0749-6419(03)00041-X

* Corresponding author. Tel.: +46-920-491284; fax: +46-920-491047.

E-mail address: karl-gustav.sundin@mt.luth.se (K.G. Sundin).1 Now at FOI, Swedish Defence Research Agency, SE-17290 Stockholm, Sweden.2 Now at Malmo University Materials Science, SE-20506 Malmo, Sweden.

1. Introduction

A technical material is usually characterised mechanically by its stress–strain curvemeasured under static conditions in a standard tensile test. In such tests the strainrate in the specimen is small, usually of the order of 10�2 s�1. However, in manytechnical applications and processes, material is deformed under conditions thatimply strain rates many orders of magnitude higher than in the standard tensile test.Examples of such processes are plastic forming and cutting. Also in unintendedsituations like collisions and impacts material is deformed at rates that can be veryhigh.A technically important situation that also involves large strains and high strainrates is the growth of a crack. A complex state of multi-axial stress and strain pre-vails in the vicinity of the crack tip. In metals the local deformation is generallyplastic and the strains are large. Even if the global loading is quasi-static, very highplastic strain-rates can be at hand close to the crack tip due to the small length scale.In the case of a propagating crack it is obvious that material is strained from zero tofracture as the crack tip passes and this process necessarily involves high strain-rates. It has been suggested that micro-cracking ahead of the main crack, leads to anirregular crack front containing small, more or less isolated regions of material,bridging the two main crack surfaces (Hogland, et al., 1972). These regions orligaments will experience very high strain rates when they are torn off as the surfacesseparate during crack propagation. The energy consumed by the plastic processes atthe crack tip will influence the propagation speed and the eventual arrest of thecrack tip. The constitutive description of the material, especially in terms of the rate-dependence, plays an important role in the modelling of this problem (Freund andHutchinson, 1985; Brickstad, 1983; Lo, 1983; Nilsson, et al., 1998).Complex, often rate dependent phenomena on micro-structural, molecular oratomic level are active during plastic deformation of a solid material. For metals thephysical explanation of the rate dependence in the constitutive relations is based onthe fact that dislocation movement is subject to rate dependent drag forces. In forexample (Barlat et al., 2002), models for dislocation behaviour and its impact onmechanical properties are presented and discussed. Unfortunately it is not yetpossible to accurately predict macroscopic material behaviour under high-rate con-ditions from dislocation theory and therefore empirical models must be used. Suchmodels must be calibrated in carefully designed experiments, through which para-meters in the models can be identified. For mild steel elastic-viscoplastic models aregenerally used for the mechanical description of high rate deformation processes.Such models have been suggested by for example Perzyna (1963), Johnson and Cook(1983, 1985) and Zerilli and Armstrong (1987). For reviews of the area see forexample Lemaitre and Chaboche (1990) and Meyers (1994). Viscoplastic con-stitutive models for high strain-rate deformation of some metals are also discussedin more recent papers, for example Liang and Khan (1999), Khan and Liang (1999)and Nemat-Nasser and Kapoor (2001). Other materials, for example polymers andwood, also exhibit rate dependent mechanical behaviour although the underlyingphysical processes obviously are different from those in metals. High-rate

562 J. Kajberg et al. / International Journal of Plasticity 20 (2004) 561–575

experiments are necessary in order to establish numerical models for transient eventsinvolving such materials. Recent reports of such experiments are for example Khanand Lopez-Pamies (2002) for a soft polymer and Widehammar (2002) for sprucewood. As long as efficient models including material parameters are not well knownamong engineers, it is probably safe to state that the demand for experimental workregarding high-rate constitutive modelling of materials will remain high.Classical experiments for determination of constitutive parameters generally uti-lise specimens designed to give a homogeneous state of stress, strain and strain ratein a volume of the tested materiel. Practical requirements regarding mechanicalattachment and strain measurement tend to demand this volume to be fairly largeand high strain rates are therefore hard to achieve in conventional testing machines.Methods like Split Hopkinson Pressure Bar (or Kolsky Bar), Rotating flywheeland Drop tower are used for high strain-rate testing. These methods generally alsoutilise a state of homogeneous stress and strain in the specimen for direct evaluationof material parameters. Relatively large specimens are therefore needed also in thesecases and high strain rates therefore require high impact velocities. In compressionstrain rates of 105 s�1 are reached in extreme experiments (Pope and Field, 1984;Gorham et al., 1992). Special versions of the split Hopkinson pressure bar have beendeveloped for tensile and shear testing and strain rates of 103 s�1 are typical in theseapplications. Comprehensive reviews of experimental methods can be found intextbooks and review papers (see for example Meyers, 1994; Blazynski, 1987;Lindholm, 1971; Field et al., 1994; Gray, 2000).In this paper a combined experimental and numerical method that does notrequire a homogeneous state of stress and strain is suggested and a pilot study isperformed. Specimens can be made small if a nonhomogeneous state is allowed andthus high strain rates can be achieved at rather low impact velocities. A smallspecimen with a non-homogenous state of stress and strain has been used also byGilat and Cheng (2000, 2002) to reach high strain rates. The method presented hereuses microscopic high-speed photography to record information regarding thedeforming geometry. Due to its non-contacting nature the suggested method isexpected to be suitable also for high temperature testing. In this first study, thegeneral principle, some preliminary test results and a sensitivity analysis arepresented. One material and one viscoplastic constitutive model are studied.

2. Experiments

A small specimen is loaded in tension by a transient force so that it is rapidlystretched to fracture. The specimen has the shape of a small ligament of materialformed by two deep, adjacent notches in a rod of rectangular (5 � 3 mm2) crosssection as shown in Fig. 1. The length, L, of the ligament is only a fraction of amillimetre (0.4 mm) in order to obtain a high strain rate when tearing the ligamentaxially. In this preliminary experiment the width, W, of the ligament, that is thedistance between the two adjacent notches, was chosen small (0.24 mm) in compar-ison with the thickness of the ligament, which is 3 mm as determined by the cross

J. Kajberg et al. / International Journal of Plasticity 20 (2004) 561–575 563

sectional dimension of the rod. With this geometry, a state of approximately planestrain will prevail through most of the thickness of the ligament. The rod in whichthe ligament is machined is 500 mm long with a T-shaped head at one end forapplication of the axial load. At a distance of 5 mm from the head the notchesforming the ligament were machined out using the method of electric wire discharge.This method allows narrow notches to be formed, using a thin wire, and thus a shortspecimen can be obtained facilitating high strain rates. A quenched and temperedA533B grade B class 1 steel with yield stress 345 MPa was chosen for this investiga-tion. This steel is used in the nuclear power industry and good understanding of itsmechanical behaviour, including the mechanics of propagating cracks, is of interest.A diagram of the experimental set-up is presented in Fig. 2. The arrangementshown in this figure is designed for tensile loading of the specimen rod butcompressive loading can easily be achieved in a similar experiment. The head of thespecimen rod is attached to one end of a loading rod with the aid of a yoke (notshown in figure). The loading rod is made of steel, it is 1.5 m long and has adiameter of 25 mm. Both rods are supported horisontally by plastic bearings withlow friction and the specimen rod is arranged along the side of the loading rod. Theloading rod is impacted axially by a projectile with a flat end-surface and thegenerated elastic wave propagates along the rod. As the wave reflects, the T-shapedhead of the specimen rod is loaded towards the left in Fig. 2. The ligament is rapidlytorn to fracture in the axial direction with an extension rate that is of the same orderas the velocity of the impacting projectile. As the material in the ligament isstretched to fracture the tensile force generates an elastic wave in the specimen rod.This tensile wave is measured by a pair of strain gauges, (KYOWA KFG-5-120-C1-11L1M2R) positioned 0.2 m from the ligament (see Fig. 1). The amplified strainsignal is recorded by a transient recorder and transferred to a computer. The

Fig. 1. The geometry of the specimen rod and the ligament (specimen). All numbers are in mm.

564 J. Kajberg et al. / International Journal of Plasticity 20 (2004) 561–575

recorded strain history, multiplied with Young’s modulus and cross-sectional area ofthe specimen rod, is taken as the force history, Fexp(t), in the ligament during therapid tearing to fracture (see Fig. 3).It should be noted here that the duration of the force history shown in Fig. 3 isabout 20 ms implying a wave length in the specimen rod of about 0.1 m. Thetensional fracture of the specimen is therefore over before the wave has reached thefar end of the 0.5 m long specimen rod. Also the positioning of the strain gauge inthe middle of the specimen rod facilitates recording of the reflected wave from thefree end without overlapping. From comparison of the primary and the reflected

Fig. 2. Experimental set-up.

Fig. 3. Measured time histories for the force Fexp(t) (solid line) and the extension U(t) (stars) evaluated

from high-speed photographs.

J. Kajberg et al. / International Journal of Plasticity 20 (2004) 561–575 565

wave it is possible to verify that disturbances from for example bending arenegligible.A high-speed camera of image-converter type (ULTRANAC FS 501) with aCCD-unit is used to record the tearing process in 15 frames with an exposure time of0.4 ms and an interval between frames of 2 ms. The camera is equipped with a longdistance microscope lens, which makes it possible to study an area of approximately1.5 � 1.5 mm2 with a stand-off distance of about 0.1 m. A high-voltage electricaldischarge from a capacitor is used to illuminate the target. The light from thedischarge is focused onto a diffuser just in front of the target area with the aid of astandard Kohler illumination system. The duration of the discharge is sufficientlylong to cover the entire tearing event but the light intensity is somewhat reducedtowards the end of the experiment and therefore the exposure time is increased alittle for the later frames.A strain gauge on the loading rod is used as triggering source. After a time delay,which is adapted to the travel time for the wave in the loading rod, the flash and thehigh-speed camera are fired. Also the transient recorder that registers the forcehistory is triggered.An example of the photographic record of an experiment is shown in Fig. 4. Thesequence covers the entire event from unloaded to completely broken ligament andthe progressive plastic deformation manifests itself through the visible extension ofthe ligament and also through the out-of-plane deformation of the plasticallystrained material at the illuminated face causing a darker appearance of theligament. The difference in appearance of the two sides of the ligament that can beobserved in Fig. 4 is caused by the oblique incidence of the illumination.The extension of the ligament was quantified from the digital high-speed photo-graphs and from the recorded force history in two steps. First, the displacement of

Fig. 4. High-speed photographs of deforming ligament. Frame interval 2 ms. Each frame coversapproximately 1.5�1.5 mm2.

566 J. Kajberg et al. / International Journal of Plasticity 20 (2004) 561–575

the two sides of the ligament, U1(t) and U2(t) (Fig. 1), were measured by tracking theposition (pixel-number in the digital picture) of the contrast change associated withthe edges of the groove in the high-speed frames. This measure was taken at a posi-tion approximately 0.8 mm from the centre of the ligament. As a second step, thedisplacement of the right end of the ligament, U2(t), was controlled using one-dimensional wave theory. This control was made because the lower contrast on thisside of the ligament introduced some uncertainty in the pixel counting. The alter-native displacement, U0

2 tð Þ, was calculated from the recorded strain history in thespecimen rod according to U0

2 tð Þ ¼ cÐ t0" �ð Þd�, where c is the one-dimensional wave

speed and "(t) is the measured tensile strain history with the time delay associatedwith the wave propagation in the specimen rod cancelled. Displacement U0

2 tð Þ cal-culated in this way agrees with the somewhat uncertain evaluation of displacementfrom the contrast change at the right edge in the photographs. That is U0

2 tð Þ � U2 tð Þand U(t)=U1(t)-U2(t) is therefore taken as the correct measure of the extension ofthe ligament. In Fig. 3, U(t) is presented at the times when high-speed photographsare taken and the time history is approximated by a spline function through thesepoints.

3. Estimation of material parameters

3.1. Inverse modelling

A high strain-rate event such as the described experiment is usually modelled withan elastic–viscoplastic material model. Such a model comprises a linear elastic partfor low stress levels and a viscoplastic part for high stress levels. Hooke’s law is usedto describe the material behaviour at low stress levels and a rate dependent model isused for the high stress levels. The rate dependent model chosen here is the onesuggested by Perzyna (1963), in which the plastic strain rates are given by

":pij ¼

32 �

�e�0� 1

� �nSij�e; �e > �0

0; �e4 �0

(ð1Þ

where �e is the effective stress according to von Mises, �0 is the yield stress and Sij isthe deviatoric stress. The model also contains two unknown parameters, namely thestrain rate sensitivity, �, and the over stress exponent, n. In this study theseparameters are estimated through so called inverse modelling which is an iterativeprocess using repeated FE-calculations with varied parameters and comparisonbetween numerical and experimental results. The method used for the inversemodelling is the subspace-searching simplex method, Subplex (Rowan, 1990). Thismethod is designed to iterate towards minimum of an objective function through so-called direct searching. That is, the value of the objective function is calculated forstepwise varied values of parameters in an iterative sequence in order to findminima. The objective function chosen in this case consists of the sum of the squaredresiduals defined as

J. Kajberg et al. / International Journal of Plasticity 20 (2004) 561–575 567

F ¼XNi¼1

f expi uð Þ � f FEi uð Þ� �2

; N ¼ 31; ð2Þ

f exp ¼F exp

�0A; f FE ¼

FFE

�0A; u ¼

U

Lð3Þ

2 exp

where A is the cross sectional area of the centre of the ligament (0.72 mm ), fi uð Þ

and f FEi uð Þ are experimentally and numerically obtained force values at 31 equallyspaced extension values in curves as those shown in Fig. 5. The curves are derivedfrom the force and extension histories corresponding to the experiment and the FE-calculation. The numerically calculated curve in Fig. 5 is the result obtained with theoptimal set of parameters. In order to ensure that a determined minimum is a globalone several different pairs of starting values for the parameters � and n have beenused in the optimisation sequence.

3.2. FE-model

An updated Lagrangian method for large deformation elastic–viscoplastic mate-rials (ABACUS/Standard 5.8) was used for the FE-calculations. The constitutivebehaviour of the material is described by Hooke’s law for the elastic part and by Eq.(1) for the plastic part. Material parameters are; Young’s modulus E=210 GPa,Poisson’s ratio �=0.3 and yield stress �0=345 MPa. As the thickness of the speci-men rod (3 mm) is much greater than the other dimensions of the ligament (L=0.4mm, W=0.24 mm) a state of plain strain is assumed through the thickness of themodelled ligament.A so called boundary layer solution is applied for limitation of the FE-model.Consider a large, linearly elastic, plane body with a ligament of width W on the

Fig. 5. Normalised force versus normalised extension. The stars indicate times when strain fields are

evaluated from the FE-simulation.

568 J. Kajberg et al. / International Journal of Plasticity 20 (2004) 561–575

symmetry-plane y=0 as shown in Fig. 6. It is loaded remotely by a force in they-direction and it can be shown (Muskhelishvili, 1953; Nilsson et al., 1998) that thedisplacement in the y-direction for points on a semicircle of radius R, is approxi-mately independent of the angle � for large values of the ratio 2R=W. As the plasticzone in the centre of the model is small this solution can be applied and the loadingof the model is therefore defined by a prescribed displacement history U tð Þ=2 in theaxial direction of all the nodes along the perimeter R. In the FE-calculations R=2.5mm is used, which is the distance from the centre of the ligament to the outer edge ofthe specimen rod. Thus 2R=W � 20 which is large enough to motivate the approx-imative boundary layer solution. U(t) is the measured extension at approximatelyR=3 as described by the dashed curve in Fig. 3. Using the extension historymeasured at radius R=3 as input at radius R in the FE-calculations implies thatelastic deformations within the semicircular region are neglected in comparison tothe large plastic deformations in the ligament. The force FFE used in the objective

Fig. 6. The modelled semicircular region, the entire mesh and a magnification of the region with the

highest element density.

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function [Eqs. (2) and (3)] is the resulting force on the circular boundary due tothe applied controlled displacement.A quasi-static approach is adopted in the FE-calculations. This is motivated bythe fact that transit times for mechanical waves through the studied structure aremuch shorter than characteristic times for the load. Therefore equilibrium of forcesis well developed and inertia forces are negligible.

3.3. Sensitivity analysis

Important for the usefulness of the method is how uncertainty in the measurementaffects the values of the estimated parameters, � and n in the constitutive model. Inorder to obtain a rough picture of this sensitivity, systematic relative uncertaintieswere introduced in the experimental force and extension histories. Uncertainty levelsof �2%, �5% for both force and extension were chosen and a new set of optimalmaterial parameters were determined for every combination of uncertainty levels.These new sets of material parameters are compared with the initial ones and thesensitivity of evaluated parameters to systematic errors in measured data canthereby be estimated.

4. Result and discussion

The experimental results from the force and extension measurements are shown inFig. 3. As mentioned above the experimental results are the basis for the FE-calcu-lation and inverse modelling. However, the use of the experimental data needs somecomments.The basis for the force measurement is one-dimensional wave propagation inthe specimen rod assuming a negligible influence from the complex stress stateclose to the ligament. Dispersion can have an influence on the propagating wavebut it is believed to be negligible because the pulse appears to have a smooth shape.Also an electromagnetic disturbance from the flash system was observed early in theforce history, but as it appeared to be repeatable it was subtracted from the forcesignal. As the specimen rod was freely supported with very low friction, reflectedwaves were observed and their similarity to the primary wave verified thatdisturbances from bending, dispersion and electromagnetic transient fields are oflittle importance.The usefulness of the measured strain history in the specimen rod as a repre-sentation of the force history in the ligament depends on the magnitude of theinertia force that is associated with the acceleration of the specimen (ligament). Avery rough and overestimated measure of the acceleration is calculated by differ-entiating the displacement history, U1(t) (splined), twice. The result is a very noisyfunction representing acceleration of the left side of the ligament which is observedto be always less than 3�106 m/s2 during the experiment. The mass of the ligamentis approximately 2�10�6 kg and it can therefore be concluded that the inertia forceon the ligament is less than 6 N at all times during the experiment. This highly

570 J. Kajberg et al. / International Journal of Plasticity 20 (2004) 561–575

overestimated value is very small compared to the measured force and therefore staticconditions are assumed in the FE-analysis for simplicity in this preliminary study.A problem with a high-speed camera of the image converter type is that relativelylarge aberrations might be introduced in the imaging. Different parts of the imagemight therefore render the object at slightly different scales. Such problems can beavoided if the camera is calibrated. At this preliminary stage these aberrations arenot taken into consideration. Such optical errors are however believed to be smalland without significant influence on measured ligament extension.The complete force and extension histories are not included in the inverse analysiswhich is obvious from a comparison of Figs. 3 and 5. Data for later times, t>12 ms,are excluded in order to avoid influence from unwanted phenomena associated withlate times in the evaluation of material parameters. The data used in the inverseanalysis are assumed to cover the viscoplastic hardening but not the necking at highstrain levels which takes place in the ligament for later times. With this restriction inthe use of force and extension histories the deformations in the FE-model do notcause so large distortions of the elements that time consuming remeshing is neces-sary. Another phenomenon which is avoided by the truncation of the data is theformation of microcracks and voids which is beyond the validity of the chosenconstitutive model.The yield stress, �0, in Eq. (1) is in general a function of plastic strain, �0("p), thusdescribing a strain-dependent yield stress. In the present calculations, a constantvalue of the yield stress was used in accordance with the original suggestion inPerzyna (1963). This rough model neglects static strain hardening and is chosen inthis pilot study for its simplicity. The value of the yield stress was obtained from astatic tension test.The inverse analysis resulted in the parameter values, �=25,000 s�1 and n=7. TheSubplex algorithm converged to the same values for all tested combinations of

Fig. 7. The objective function, , in the neighbourhood of the determined minimum.

J. Kajberg et al. / International Journal of Plasticity 20 (2004) 561–575 571

starting parameters. These were chosen in a wide range from 0.1 to 10 times theestimated parameter values. In Fig. 7 the objective function, , according to Eqs. (2)and (3) is plotted as a 3D-surface with its corresponding level curves. It is observedthat, at least in the area covered by the diagram, the objective function appears tohave a single minimum.The sensitivity analysis shows substantial changes in the optimal parameter valueswhen uncertainties of up to 5% are introduced for the measured quantities, Fexp(t)and U(t). Fig. 8 illustrates the relative changes of the evaluated optimal parametersas functions of systematic relative errors in measured quantities. It is observed that �is more sensitive than n to systematic errors and that � is more sensitive to errors inFexp than it is to errors in U. These sensitivities indicate that accuracy in measuredquantities is highly important for the reliability of estimated material parameters.

Fig. 8. Relative changes of the optimal parameters, � and n, when uncertainties are introduced for themeasured quantities Fexp(t) and U(t).

572 J. Kajberg et al. / International Journal of Plasticity 20 (2004) 561–575

An interesting quantity to estimate is the strain rate in the ligament. In Fig. 3where extension vs. time is plotted, the maximum extension rate can be estimated to8 m/s for the studied time period (t<12 ms). This rate divided by the ligament length,L gives a rough estimation of the strain rate of 2�104 s�1. Another way to estimatethe strain rate is to use the results from the FE-calculation with the optimum mate-rial parameters. Fig. 9 shows the distribution of the strain rate in the y-direction atthree different instants (9.6, 10.8, 12.0 ms) indicated with stars in the fFE(u) curve inFig. 5. These fields show that the highest strain rate is obtained in the middle of theligament and it reaches about 7�104 s�1 at the third and latest instant. This strainrate is about three times higher than the rough estimation and this is expected forthe nonhomogeneous strain field caused by the ligament geometry.

5. Conclusions

The presented method is aimed as a tool for determination of constitutive para-meters under high strain-rate conditions and the basis for the method is iterativenumerical simulation of experimental results or so called inverse modelling. Thesepreliminary results indicate that the mechanical behaviour of a small specimen canbe observed and evaluated at strain rates up to 104–105 s�1.A series of high-speed photographs was taken of the small ligament area duringthe deformation process. The quality of the frames is sufficient for investigation ofthe deformation and fracture process and evaluated data can be used in inverse

Fig. 9. Strain-rate distributions at three different instants (9.6, 10.8, 12.0 ms) during the deformationprocess. The instants are marked with stars in Fig. 5.

J. Kajberg et al. / International Journal of Plasticity 20 (2004) 561–575 573

modelling. However, only one deformation quantity is measured in this preliminarystudy making results vulnerable to experimental errors.Material parameters in Perzyna’s viscoplastic model were estimated throughinverse modelling and a sensitivity analysis was performed, that quantifies therelation between accuracy in evaluated parameters and in measured data.Simulation of the experiment using estimated optimal parameters showed goodagreement with experimental result.We conclude the following:

. The experimental method is promising but needs some improvement regardingresolution in the optical measurement.

. Numerical simulation of experiments is necessary and methods like Subplexfacilitates the extraction of constitutive parameters.

. Microscopic high-speed photography is a powerful method to study a transientdeformation process in a small specimen and with optical methods like Moireand speckle methods more information regarding the deformation field couldbe recorded.

Acknowledgements

The authors are grateful to Dr. Bengt Wikman and Dr. Greger Bergman for theirvaluable guidance with the implementation of the Subplex algorithm. TheULTRANAC camera purchase was possible due to a grant from the SwedishCouncil for Planning and Coordination of Research (FRN).

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